This document is a thesis submitted by Bin Liu to the Graduate College of The University of Texas Rio Grande Valley in partial fulfillment of the requirements for a Master of Science degree in Mechanical Engineering. The thesis investigates the nonlinear dynamics of electrostatically actuated carbon nanotube nanotweezers, including primary and parametric resonance responses. Analytical and numerical methods are used, including the Method of Multiple Scales and a Reduced Order Model approach. Effects of various system parameters on the frequency and voltage responses are analyzed.

1. NONLINEAR DYNAMICS OF ELECTROSTATICALLY
ACTUATED NANOTWEEZERS
A Thesis
by
BIN LIU
Submitted to the Graduate College of
The University of Texas Rio Grande Valley
In partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE
May 2016
Major Subject: Mechanical Engineering

3. NONLINEAR DYNAMICS OF ELECTROSTATICALLY
ACTUATED NANOTWEEZERS
A Thesis
by
BIN LIU
COMMITTEE MEMBERS
Dr. Dumitru Caruntu
Chair of Committee
Dr. Mircea Chipara
Committee Member
Dr. Mohammad Azarbayejani
Committee Member
May 2016

5. Copyright 2016 Bin Liu
All Rights Reserved

7. iii
ABSTRACT
Bin Liu, Nonlinear Dynamics of Electrostatically Actuated Nanotweezers. Master of Science
(MS), May, 2016, 73 pp., 6 tables, 50 figures, references, 35 titles.
The amplitude-frequency and amplitude-voltage responses of electrostatically actuated
carbon nanotube (CNT) nanotweezers device system is investigated. Soft alternate current (AC)
near-half natural frequency of CNTs leads to primary resonance, while AC near natural
frequency of CNTs leads to parametric resonance. Both resonances are reported. Two methods
are used, namely the Method of Multiple Scales (MMS) to obtain analytical approximate
solutions, and the Reduced Order Model (ROM) method to numerically simulate the behavior of
the system. Effects of van der Waals molecular forces, electrostatic forces, damping forces,
excitation frequency, and excitation voltage on frequency and voltage responses are reported.

9. iv
DEDICATION
The accomplishment of my graduation is based on my family’s numerous sacrifices and
great support from my thesis committee chair Dr. Caruntu, the chairman of mechanical
engineering department, Dr. Freeman, graduate advisor, Dr. Vasquez, and especially great help
from fellow colleague Christian Reyes. To my father, Junmin, my mother, Sufang, my brothers
Yan and Jie, and particularly my wife, Jing, who quit her job to give me a hand on my daily life
so that I can focus on my thesis. Thank you for all your valuable encouragement and inspiration
that pushed me to achieve my degree.

11. v
ACKNOWLEDGEMENTS
I would like to acknowledge my thesis committee chair, Dr. Caruntu, for all his great
support including but not limited to valuable advices, encouragement, and countless days. Being
my research advisor, he helped me develop and accomplish an in-depth and high quality thesis. I
also greatly appreciate the help I received from my thesis committee members Dr. Mircea
Chipara and Dr. Mohammad Azarbayejani. Thanks for their taking part in the committee as well
as their valuable suggestions to improve my thesis.

13. vi
TABLE OF CONTENTS
Page
ABSTRACT...................................................................................................................................iii
DEDICATION............................................................................................................................... iv
ACKNOWLEDGEMENTS............................................................................................................ v
TABLE OF CONTENTS............................................................................................................... vi
LIST OF TABLES.......................................................................................................................viii
LIST OF FIGURES ....................................................................................................................... ix
CHAPTER I. INTRODUCTION.................................................................................................... 1
CHAPTER II. SYSTEM MODEL.................................................................................................. 7
2.1 Dynamic model ..................................................................................................................... 8
2.2 Dimensionless equation......................................................................................................... 9
2.3 Taylor expansion coefficients ............................................................................................. 10
CHAPTER III. METHOD OF MULTIPLE SCALES DIRECT APPROACH............................ 11
3.1. Method of Multiple Scales for Primary Resonance (AC near Half Natural Frequency). .. 11
3.2. Method of Multiple Scales parametric resonance (AC near Natural Frequency).............. 16
3.3. Frequency response for primary and parametric resonances using MMS ......................... 17
CHAPTER IV. REDUCED ORDER MODEL FOR ELECTROSTATIC ACTUATION .......... 20
CHAPTER V. HALF NATURAL FREQUENCY RESONANCE RESPONSE......................... 26
5.1 Amplitude-Frequency Response for Primary case.............................................................. 26
5.2. Voltage-Amplitude Response for Primary Resonance....................................................... 38
CHAPTER VI. NATURAL FREQUENCY RESONANCE RESPONSE................................... 47

14. vii
6.1 Amplitude-Frequency Response for Parametric resonance. ............................................... 47
6.2. Voltage-Amplitude Response for Parametric Resonance.................................................. 57
CHAPTER VII. DISCUSSION AND CONCLUSIONS ............................................................. 66
7.1. Summary ............................................................................................................................ 66
7.2. Future Work ....................................................................................................................... 68
REFERENCES ............................................................................................................................. 70
BIOGRAPHICAL SKETCH........................................................................................................ 73

15. viii
LIST OF TABLES
Page
Table 1: Coefficients of Taylor expansion in numerator...............................................................10
Table 2: Natural frequencies and mode shape coefficients for CNTs. ..........................................13
Table 3: System Constants.............................................................................................................17
Table 4: Dimensionless Parameters of the System........................................................................17
Table 5: Dimensionless System Parameters ..................................................................................18
Table 6: Coefficients of Taylor expansions in the denominator....................................................20

17. ix
LIST OF FIGURES
Page
Figure 1: Nanotweezers model schematic diagram ........................................................................ 7
Figure 2: Amplitude frequency response of primary resonance using MMS. AC frequency
near half fundamental natural frequency 1=3.5165. = 0.005, = 0.15, b*=0.001. ................. 18
Figure 3: Amplitude frequency response of parametric resonance using MMS. AC frequency
near fundamental natural frequency 1=3.5165. = 0.005, = 0.15, b*=0.001. ........................ 19
Figure 4: Three terms ROM time response................................................................................... 25
Figure 5: Amplitude frequency response of primary resonance using MMS and 3T ROM. AC
frequency near half fundamental natural frequency 1=3.5165. = 0.005, = 0.15, b*=0.001. 27
Figure 6: Phase frequency response of primary resonance using MMS. AC frequency near
half fundamental natural frequency 1=3.5165. = 0.005, = 0.15, b*=0.001,1=3.5165....... 28
Figure 7: Phase plane for three steady-state points P1, P2, and P3 in Fig. 5. AC frequency
near half natural frequency 1=3.5165. = 0.005, = 0.15, b* = 0.001. ................................... 29
Figure 8: Time response using 3T ROM. AC frequency near half fundamental natural
frequency 1=3.5165. U0 = 0.10, = 0.15, = 0.005, = -0.03, b* = 0.001............................. 30
Figure 9: Time response using 3T ROM. AC frequency near half fundamental natural
frequency 1=3.5165. U0 = 0.10, =0.15, = 0.005, = -0.008, b*=0.001................................ 31
Figure 10: Zoom in of Fig. 5. AC frequency near half fundamental natural frequency
1=3.5165. = 0.005, = 0.15, b*=0.001. .................................................................................. 31
Figure 11: Time response using 3T ROM. AC frequency near half fundamental natural
frequency 1=3.5165. U0 = 0.45, =0.15, = 0.005, = -0.008, b*=0.001................................ 32
Figure 12: Time response using 3T ROM. AC frequency near half fundamental natural
frequency 1=3.5165. U0 = 0.2, = 0.005, = 0.15, b* = 0.001, = -0.0055......................... 33
Figure 13: Time response using 3T ROM. AC frequency near half fundamental natural
frequency 1=3.5165. U0 = 0.1, = 0.005, = 0.15, b* = 0.001, = -0.03............................. 34

18. x
Figure 14: Effect of voltage parameter  on the amplitude frequency response using 3T
ROM. AC frequency near half fundamental natural frequency 1=3.5165. = 0.005, b* =
0.001.............................................................................................................................................. 35
Figure 15: Effect of damping parameter b* on the amplitude frequency response using 3T
ROM. AC frequency near half fundamental natural frequency 1=3.5165. = 0.005, =0.15.. 36
Figure 16: Effect of van der Waals parameter  on the amplitude frequency response using
3T ROM. AC frequency near half fundamental natural frequency 1=3.5165. b*= 0.001,
=0.15. ......................................................................................................................................... 37
Figure 17: Convergence of the amplitude frequency response showing MMS, two terms (2T)
ROM, and three terms (3T) ROM. AC frequency near half fundamental natural frequency
1=3.5165. = 0.005, = 0.15, b* = 0.001. ............................................................................... 38
Figure 18: Amplitude voltage response of primary resonance using MMS and 3T ROM. AC
frequency near half fundamental natural frequency 1=3.5165. = 0.005, = -0.004,
b*=0.001. ...................................................................................................................................... 39
Figure 19: Phase plane of three steady-state points P4, P5, and P6 in Fig. 19. AC frequency
near half natural frequency 1=3.5165. = 0.005, = 0.15, b* = 0.001. ................................... 40
Figure 20: Time response using 3T ROM. AC frequency near half fundamental natural
frequency 1=3.5165. U0 = 0.1, =0.01, = 0.005, = -0.004, b*=0.001.................................. 41
Figure 21: Time response using 3T ROM. AC frequency near half fundamental natural
frequency 1=3.5165. U0 = 0.20, =0.04, = 0.005, = -0.004, b*=0.001................................ 42
Figure 22: Time response using 3T ROM. AC frequency near half fundamental natural
frequency 1=3.5165. U0 =0.30, =0.04, = 0.005, = -0.004, b*=0.001................................. 42
Figure 23: Time response using 3T ROM. AC frequency near half fundamental natural
frequency 1=3.5165. U0 = 0.1, =0.05, = 0.005, = -0.004, b*=0.001.................................. 43
Figure 24: Time response using 3T ROM. AC frequency near half fundamental natural
frequency 1=3.5165. U0 = 0.1, =0.07, = 0.005, = -0.004, b*=0.001.................................. 44
Figure 25: Effect of detuning frequency parameter on the amplitude frequency response
using 3T ROM. AC frequency near half fundamental natural frequency 1=3.5165. =0.005,
b*
=0.001........................................................................................................................................ 44
Figure 26: Effect of damping parameter b*on the amplitude frequency response using 3T
ROM. AC frequency near half fundamental natural frequency 1=3.5165. = 0.005,
b*=0.001........................................................................................................................................ 46

19. xi
Figure 27: Effect of damping parameter b*on the amplitude frequency response using 3T
ROM. AC frequency near half fundamental natural frequency 1=3.5165.  = -0.004,
b*=0.001........................................................................................................................................ 46
Figure 28: Amplitude frequency response of primary resonance using MMS and 3T ROM.
AC frequency near fundamental natural frequency 1=3.5165. = 0.005, = 0.15, b*=0.001.. 48
Figure 29: Phase frequency response of primary resonance using MMS. AC frequency near
fundamental natural frequency 1=3.5165. = 0.005, = 0.15, b*=0.001. ............................... 49
Figure 30: Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0.35. =0.15, = 0.005, = -0.0125................................................................ 50
Figure 31: Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0.05. =0.15, = 0.005, = -0.015.................................................................. 50
Figure 32: Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0. =0.15, = 0.005, = -0.006....................................................................... 51
Figure 33: Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0.4. =0.15, = 0.005, = -0.01...................................................................... 52
Figure 34: Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0.1. =0.15, = 0.005, = -0.003.................................................................... 53
Figure 35: Effect of voltage parameter  on the amplitude frequency response using 3T
ROM. AC frequency near fundamental natural frequency 1=3.5165. = 0.005, b*=0.001...... 54
Figure 36: Effect of damping parameter b*on the amplitude frequency response using 3T
ROM. AC frequency near fundamental natural frequency 1=3.5165. = 0.005, =0.15......... 55
Figure 37: Effect of damping parameter on the amplitude frequency response using 3T
ROM. AC frequency near fundamental natural frequency 1=3.5165. b*
= 0.001, =0.15......... 55
Figure 38: Convergence of the amplitude frequency response showing MMS, two terms
(2T) ROM, and three terms (3T) ROM AC frequency near fundamental natural frequency
=3.5165. = 0.005, = 0.15, b*=0.001. ................................................................................. 56
Figure 39: Amplitude voltage response of primary resonance using MMS and 3T ROM. AC
frequency near fundamental natural frequency 1=3.5165. = 0.005, = -0.006, b*
=0.001..... 58
Figure 40: Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0.2, = 0.05, = 0.005, = -0.006, b* = 0.001............................................... 59

20. xii
Figure 41: Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0.1. =0.12, = 0.005, = -0.006, b*=0.001................................................... 59
Figure 42: Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0.1. =0.08, = 0.005, = -0.006, b*=0.001................................................... 60
Figure 43: Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0.30, =0.15, = 0.005, = -0.006, b*=0.001................................................. 61
Figure 44: Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0.3. =0.32, = 0.005, = -0.006, b*=0.001................................................... 62
Figure 45: Effect of detuning frequency parameter on the amplitude frequency response
using 3T ROM. AC frequency near fundamental natural frequency 1=3.5165.= 0.005,
b*=0.001........................................................................................................................................ 62
Figure 46: Zoom in of Fig. 35 for amplitude frequency response showing MMS, two terms
(2T) ROM, and three terms (3T) ROM. AC frequency near fundamental natural frequency
1=3.5165. = 0.005, = 0.15, b*=0.001. ................................................................................. 63
Figure 47: Effect of detuning frequency parameter b*on the amplitude frequency response
using 3T ROM. AC frequency near fundamental natural frequency 1=3.5165. = 0.005,
b*=0.001........................................................................................................................................ 64
Figure 48: Effect of detuning frequency parameter on the amplitude frequency response
using 3T ROM. AC frequency near fundamental natural frequency 1=3.5165.= -0.006,
b*=0.001........................................................................................................................................ 65
Figure 49: Convergence of the amplitude voltage response showing MMS, two terms (2T)
ROM, and three terms (3T) ROM. AC frequency near fundamental natural frequency
1=3.5165.= 0.005, = 0.15, b*=0.001. .................................................................................. 65
Figure 50: Distance between nanotweezers with DC voltage. Pull-in voltage............................. 68

21. 1
CHAPTER I
INTRODUCTION
Since their discovery in 1991 by Iijima [1], carbon nanotubes (CNT) have been used in
various applications due to their benefits of higher density, smaller size, lower power
consumption and other excellent mechanical, electrical and chemical properties. They are
utilized for various devices, such as sensors, electrostatic switches and nanorelays. Among these
nanodevices, nanotweezers, which consist of two carbon nanotube arms, have been drawing a lot
of attention.
From 1981, IBM researchers, Bining and Rohrer, developed the scanning tunneling
microscope (STM) [2], making it possible to work at nano scales for the first time. The STM
scans objects by applying a tunneling current to connect an ultra-sharp tip and target, while also
generating a surface topography in a tri-dimensional scale. However it is limited to non-
conducting objects. To eliminate such limitation, in 1986 Binning et al. invented the atomic force
microscope (AMF) [3], which replaces tunneling currents with force interaction between the
sharp tip and object to generate topography. A common AFM tip attaches two CNTs on two
sides of the tip with a focused ion beam separating the CNT’s, while aluminum lines patterned
on the AFM tip cantilever beams allow the driving voltage current to get through. Since then,
nanotweezer technology has become one of the main applications based on scanning probe
microscopy (SPM) tip. However, SPMs have another limitation to manipulate samples, including

22. 2
measuring their physical properties causing their single probe tips. Obviously, two probes
couldmanipulate samples, which is now known as nanotweezers. The following researchers
utilized nanotweezer in the fields of SPM [4, 5]. Eigler and Schweizer [6] showed precisely the
position of single atoms by using the sharp tip of a scanning tunneling microscope, and
discovered nanotweezers have strong potential in the field of nanomanipulation and
measurement. It can be used for switching of single atoms, manipulating nanomaterials as well
as mapping structures. Also, with the advantage of high aspect ratio, nanotubes are a good
candidate for manipulating biological entities.
Various materials have attempted to be utilized to fabricate nanotweezers, such as GaN
[7], silicon [8] using microfabrication techniques. Zero-dimensional quantum, one-dimensional
and two-dimensional quantum, structures have been used to attempt synthesizing nanotweezers
[7]. Kim and Lieber [9] successfully used nanometer-diameter carbon nanotubes to create robust
nanotweezers. Since carbon nanotubes are ideal materials as they have a remarkable mechanical
toughness and electrical conductivity, their size can be as small as 1 nm. By applying increasing
bias voltages from 0 to 8.3V to the electrodes, the electromechanical response of nanotube
nanotweezers was clearly demonstrated. As the applied voltage was increased, the tweezer arms
bent closer to each other from their initial position, and removing voltage caused the tweezer
arms to returns back to their relaxed position. This procedure was repeated more than 10 times to
show the response is elastic. Various attempts at CNT-based nanotweezers have been
investigated [10-12].
Nanotube nanotweezers are composed of two flexible nanotubes, which are referred to as
arms, with one end attached to independent electrodes. The free end of nanotubes close and open
with the voltage applied to the electrodes varying, the displacement between two arms referred

23. 3
as tweezing range. In order to achieve a wanted equilibrium [13] position, a direct current (DC)
voltage impulse is used to balance the restoring force within the flexible nanotubes. Alternating
current (AC) will be an additional impulse on the DC voltage to generate vibration for the
system, which are known as resonators [14]. With the voltage increasing from 0 V slowly, the
electrostatic force gradually overcomes the elastic restoring force of arms, and the equilibrium
position converges closer and closer to the center line. Beyond a critical value, the stability of the
equilibrium position is lost, the system becomes unstable and the two arms collapse on each
other. This characteristic is known as the pull-in phenomenon [15], where the corresponding
voltage is called pull-in voltage. Thus, the pull-in phenomenon limited the tweeze range. As
electrostatic nanotubes are quite sensitive, the presence of coulomb electrostatic forces, van de
Waals molecular attractions and damping forces which are caused by the combined inherent
properties, air influence and temperature facts must be taken into consideration to avoid this pull-
in phenomenon causing failure of the system [16].
In order to get a better understanding of nanotweezers, different methods have been used
to investigate the dynamic behavior, mostly employing molecular mechanics or dynamics
simulation to study the grapheme interaction of the tweezers [17]. The dynamics simulation
methods showed how the linear model results matched the experimental data closely, while
molecular mechanics can accurately simulate the physical properties and chemical properties of
the arms in nano level. However, these two methods are limited by the linear model considering
nanotubes mostly show nonlinear behavior. Lumper parameter model [18, 19] and nanoscale
continuum models [20-21] are improved methods. By applying a Semi-lumped Parameter
Model, the physics of nanotube arms can be directly displayed. Compared with experimental
results, the Lumper parameter model can easily display the physical behavior of the

24. 4
nanotweezers, but failed to provide precise and reliable results. Continuum models simplified the
influence of physical parameters into closed formed simple formulas, making parametric study
possible. It successfully solved time-consuming problems by reducing the analyzing time needed
and providing more precise results compared with Lumped parameter modes, but they did not
pay attention to electrostatic forces, which have significant influence to those sensible nanoscale
structures [22].
Amplitude-frequency response gives the relationship between the steady-state amplitudes
of the CNT and the frequencies of actuation. Effects of different parameters on the response of
the system are reported. The frequency response obtained using the Method of Multiple Scales
(MMS) show the frequencies corresponding to the bifurcation points, which are points where the
stability of the nanotweezers device changes. Caruntu and Luo [23] reported the amplitude
frequency response carbon nanotube (CNT) cantilever above a parallel plate and under the
influence of electrostatic and van der Waals forces. They reported CNT cantilevers primary
resonance. Similarly, Caruntu and Martinez [24], and Caruntu and Knecht, [25] presented the
influence by fringing effect and Casimir effects on the response of MEMS cantilever resonator
under electrostatic actuation.
Voltage-amplitude response is another bifurcation diagram similar to the frequency-
amplitude response. Voltage-amplitude response illustrates the relationship between the
oscillation amplitudes of the nanotubes and the actuating voltage. In the case of this response the
detuning frequency is held constant. This bifurcation diagram predicts the changes in stability of
the structure. Chen, X. Q. and Saito, T. [26] investigate the influence of voltage to the deflection
of nanotweezers. Caruntu et al. [23, 27] reported the voltage-amplitude responses of MEMS
cantilevers as well as MEMS plates.

25. 5
Casimir force and van der Waal force are becoming significant and playing a
fundamental role when investigating at nano-scale. They describe the same phenomenon but at
different size scales. Van der Waal force (or van der Waal’s interaction) is a general term to
define an attraction or repulsive force between two flexible bodies, tubes, plates or cantilever, or
parts of the same molecule. Casimir forces are physical forces generated in electromagnetic
fields due to quantum vacuum fluctuation (quantum field theory [28]); it is small attractive force
acting between two closed parallel, uncharged conducting nanotubes. At nanoscale, with the gap
between electrodes nanotube decreasing, typically below 20nm, the intermolecular force (van de
Waals attraction) has a significant impact on the system. For cases typically above 20nm, the gap
is longer than wavelength of the nanotube surface and the quantum fluctuation becomes
significant, which means Casimir forces become more important.
Damping force is another important factor influencing the Micro/ Nano systems behavior.
The damping effecting the nano system is a combination of surrounding fluid and the intrinsic
damping. Previous research [29] on Micro/Nano resonators showed that the gas damping is
significant. Therefore, to predict the behavior of nanotweezers one must take into consideration
the gas damping. Weibin and Turner [30] used both experimental and analytical methods and
reported the damping ratio to air pressure for Micro / Nano resonators.
In this present investigation, the method of multiple scales (MMS) will be used first to
investigate and discuss the behavior of nanotweezers under the influence of electrostatic force,
van de Wall force, as well as damping forces. The AC frequency of actuation is considered in
two cases, near half natural frequency, and near natural frequency of the nanotubes. Another
method, Reduced Order Model (ROM) [25], is used to numerically investigate the behavior of
nanotweezer system. Amplitudes with different parameter values will be compared with each

26. 6
other to show effect of those parameters on the nanotweezers response. The effects of
electrostatic force and van der Waal force on the response are investigated as well. The results of
the two methods MMS and ROM are compared. They show an excellence agreement for
amplitudes less than 0.3 of the gap. One should mention that MMS is valid for small amplitudes,
only.

27. 7
CHAPTER II
SYSTEM MODEL
The nanotweezers structure is composed of two cantilever conductor carbon nanotubes
which are acting as branches, connected by a soft alternate current (AC) voltage, Fig. 1. Between
them is nonconductor air, providing resistance considered as damping. When the given current
arises to critical values, those two beams are pushed towards each other until they make contact.
This phenomenon is called pull-in. The gap between two beams is denoted by g .
Figure 1. Nanotweezers model schematic diagram.

28. 8
2.1 Dynamic model
The partial differential equation of carbon nanotweezers electrostatically actuatedis given
as     d
F
v
F
e
F
x
w
xEI
xt
w
xA 












2
2
2
2
2
2
 (1)
Where
vF ,
eF and
dF are the van der Waals force, electrostatic force and damping force,
respectively [31].  is the density of nanotubes, E is the Young modulus, I is the cross-section
moment of inertia of the nanotube. These forces are given by the following equations with the
assumption that the radius of nanotweezers is much smaller than the gap between the nanotubes
2/52/3
2
)2(2128
3
wgr
H
Fv



(2)
t
w
cFd


 (3)
2
2
2
1
2
ln
2
1


































g
w
r
g
g
w
g
v
F o
e

(4)
The AC voltage is as follows
t cos0 (5)
In equations (2-5), g is the gap (initial distance between nanotweezers), w is deflection
of nanotweezers, v is voltage amplitude, c is damping constant. h is thickness of nanotubes, H is
the Hamaker constant [32], εo is the permitivity of free space, r is the radius, 0 is the AC
voltage amplitude and  is the AC frequency.

29. 9
2.2 Dimensionless equation
The following variables were used to non-dimensionalize the equation:
g
w
u  ,
l
x
z  ,
o
o
A
EI
l
t

 2
 (6)
Whereu , z and  are dimensionless deflection, longitudinal coordinate, and time,
respectively. The dimensionless partial differential equation of motion becomes,
 
























 w
b
u
r
g
u
uz
w
t
w *
22/54
4
2
2
21
2
ln)21(
*2cos
)21( (7)
Where the dimensionless van der Waals, voltage, and damping parameters *
,, b ,
respectively, are given by
oEIgr
lHh
2/72/3
42
2128
3
 (8)
2
2
4
0
o
o
v
EIg
l
  (9)
oo AEI
cl
b
/
*
2
 (10)
And the dimensionless frequency of actuation and natural frequency ,*
 , respectively,
are as follows
o
o
A
EI
l 2
* 
 (11)
o
o
A
EI
l 

 2
 (12)

30. 10
One should mention that , are the corresponding dimensional parameters.
2.3 Taylor expansion coefficients
Using Taylor expansion to expand w at the point 0w , Eq. (12) becomes


 *cos
2
2
00
2
4
*
2









 
k
n
k
k
k
n
k
k ww
z
ww
b
t
w (n = 0, 1, 2 ….) (13)
Where *
b is a small damping parameter, is a small voltage parameter,  is a small van
der Waals parameter,  is a bookkeeping device, w is the deflection of nanotweezers, *
 is the
actuation frequency, k are coefficient of electrostatic in Taylor expansions, k are coefficient of
van der Waals force in Taylor expansions, n is the number of Taylor expansion terms. The
coefficients of Eq. (13) are given in Table 1.
Table 1. Coefficients of Taylor expansion in numerator.
Symbol Value Symbol Value
0 0.0735 0 1
1 0.2267 1 5
2 0.5978 2 17.5
3 1.7483 3 52.5
4 3.5259 4 144.375
5 8.2144 5 375.375

31. 11
CHAPTER III
METHOD OF MULTIPLE SCALES DIRECT APPROACH
3.1. Method of Multiple Scales for Primary Resonance (AC near Half Natural Frequency).
The first approach to modeling the behavior of the electrostatically actuated clamped-free
nanotweezer is using the method of multiple scales (MMS). This analytical perturbation method
provides an understanding of the behavior of a system within a localized range of frequencies.
MMS is used to find the response of the system such as amplitude-frequency response, i.e the
relationship of the amplitude of vibration and the excitation frequency. MMS can also be used to
find the voltage-amplitude response, i.e the relationship between the amplitude of vibration and
the excitation voltage. In the MMS model, the assumption is that the system is weakly
nonlinear. MMS transforms the complex nonlinear model of the system into several simpler
linear equations. Two scales are considered for this model, the fast scale T0 and the slow scale T1
  1, TTo (14)
In order to use MMS, the electrostatic and van der Waals terms of the equation must be
expanded using Taylor expansion. The bookkeeping device  is used on every term to mark the
small terms in the system. The system is assumed weakly nonlinear as well. The deflection w
depends on both fast and slow time scale 10 , TT . One can write the first-order expansion as
1www o  (15)

32. 12
Where ow and 1w are the zero-order and first-order approximation solutions of the
equation (13), respectively. The time derivative can be expressed in terms of derivatives with
respect to the fast and slow scales as:
1DD
d
d
o 

 .1,0,  i
dT
d
D
i
i (16)
Substituting equations (14), (15), (16) back into equation (13), one obtains:
)(cos
)()())(2(
3
0
*2
3
0
10
4
1010
*
10
2
1
2
10
2
0
 





k
k
k
k
k w
ww
z
wwDDbwwDDDD

 ）（
(17)
Considering 2
 <<1, we disregard terms having the coefficient of 2
 , then, by equating
the coefficient of 0
 and 1
 , one arrives at the following two problems called zero-order and first-
order, respectively:
:0
 04
0
4
0
2
0 



z
w
wD (18)
:1
 )(cos*2
3
0
*2
0
3
0
0000104
1
4
1
2
0  




k
k
k
k
k
k wwwDbwDD
z
w
wD  (19)
The solution of the boundary value problem associated to the zero-order problem is given
by the form:
    00
110 )( TiTi
k
kk
eTAeTAzw 
 
 (20)
))sin()(sinh()cos()cosh()( xxcxxz kkkkk   (21)
Where k are the mode shapes, A is a complex amplitude (depending on the slow time
scales 1T ) to be determined. The first five dimensionless natural frequencies and mode shape

33. 13
coefficients are given in Table2. The first frequency in this table is called fundamental natural
frequency.
Table 2. Natural frequencies and mode shape coefficients for CNTs [25].
k=1 k=2 k=3 k=4 k=5
k 3.51602 22.0345 61.70102 1200.91202 199.85929
kc -0.734 -1.0185 -0.9992 -1.00003 -1.00000
Substituting Eq. (20) into Eq. (19) gives:
)]33(
)()([
)]33()2
()()[2(
4
))(())((2-
2233
3
3
2222
210
22333
3
22
222
210
22
*
4
1
4
1
2
0
**
okokok
okokokok
okokokok
okokokoo
okokokok
TiTiTi
k
TiTi
k
TiTi
k
TiTiTi
k
Ti
Ti
k
TiTi
k
TiTi
kk
TiTi
kk
TiTi
eAAeAAeA
eAeAeAAe
eAAeAAeAeAAA
eAeAAeee
ieAAebieAAe
z
w
wD























(22)
Multiplying the right-hand side of Eq. (22) by k and integrating from 0 to 1, mark
nkkk
n
k gdz   , , results gives as,
0)]33()2(
)([)]33(
)2()([
)2(
4
))(())((2-
2233
33
2222
22
1100
2233
33
2222
221100
22
1
*
1
**








okokokokok
okokokokok
okokokok
oookokokok
TiTiTi
kk
TiTi
kk
TiTi
kkkk
TiTiTi
kk
TiTi
kk
TiTi
kkkk
TiTi
kkk
TiTi
kkk
TiTi
eAAeAAeAgeAAAeAg
eAAeggeAAeAAeAg
eAAAeAgeAAegg
eegieAAebgieAAe









(23)
Solvability condition for Eq. (23) requires the right-hand side of this equation to be
orthogonal to the solutions of the homogeneous equation. Therefore, coefficients of the secular
terms ( okTi
e  ) are summed and set equal zero:
 )3()3(
2
'2- 2
3311
2
33111
*
1
okTi
kkkkkkkkkkkkkk eAAgAgAAgggAibgiA 




34. 14
+“Other secular terms”=0 (24)
The given terms in Eq. (24) come from all terms with ( okTi
e 
) in Eq. (22), except the ones
multiplied by oo TiTi
ee
**
22 
 , and given in Eq. (23), which will contribute to the “Other secular
terms.” Since we are interested in the behavior of the CNTs nanotweezers primary resonance
with a softening AC (alternate current) near its half fundamental natural frequency, the excitation
frequency *
 becomes



2
* 1
(25)
Where is the detuning parameter that varies the frequencies of excitation. Hence:
10
*
2
TT k


 (26)
Substituting Eq. (27) into Eq. (24), the “other secular terms” in the Eq. (24) are found.
Therefore the equation of secular terms becomes:
0)3(
42
42
3
2
'2-
2
3311
2
0
2
22
22
22
2
33111
*
1
11
1




AAgAgeAeAg
eAgAAgAggAibgiA
kkkk
iTiTi
kk
Tii
kkkkkkkkkkkk













(27)
Express the complex amplitude A in polar form:
i
aeA
2
1
 (28)
Where a is the real amplitude,  is the phase. Substituting Eqs. (26) and (28) into Eq.
(27), it results:
0
8
3
2
1
4816
16
3
42
1
)
2
1
2
1
(2-
3
3311
2
0
22
22
222
22
3
3
3111
*
1
111








 










i
kk
i
kk
TiTi
kk
Tii
kk
i
kkkk
i
kkk
i
kkk
ii
eagaegeeageag
egagaegiaebgiiaeea
(29)

35. 15
Dividing by i
e , Eq. (29) becomes
0
8
3
2
1
4816
16
3
42
1
)
2
1
2
1
(2-
3
3311
2
0
22
22
22
22
3
3
3111
*
1
111









agageeageag
gaaggaibgiaia
kkkk
iTiiTi
kk
Tii
kk
kkkkkkkkkk










(30)
Introducing a new variable phase difference between actuation and reaction frequency ,
  12 T , hence   2 (31)
And using Euler's formula, the equation of real terms of Eq. (30) is given by
0cos
4
cos
16
3
8
3
2
1
16
3
4
0
2
22
3
33113
3
3111









ag
agaggaagga
kk
kkkkkkkkkkk
(32)
And the equation of imaginary terms of Eq. (30) by
0sin
4
sin
162
1
- 0
2
221
*
1  



 aggabgia kkkkkkkk (33)
Eqs. (32) and (33) lead the frequency - amplitude and phase-amplitude differential given
by:






 





sin
4
sin
162
11
0
2
221
*
1
aggab
g
a kkkkk
kkk
(34)
























cos
4
cos
16
3
8
3
2
1
16
3
4
2
1
0
2
22
3
33113
3
311
1
ag
agaggaag
ga
kk
kkkkkkkk
kkk
(35)
The steady-state solution correspond to 0 a and is given by:








 




sin
4
sin
16
2
0
2
22
1
*
ag
gb
a kk
kkk
(36)

36. 16






















cos
8
cos
32
3
16
3
4
1
32
3
8
0
2
22
3
33113
3
311
ag
agaggaag
kk
kkkkkkkk
(37)
3.2. Method of Multiple Scales parametric resonance (AC near Natural Frequency)
MMS parametric response investigated the nanotweezers system with a softening AC,
compared with primary resonance response, the frequencies considerate ranges around its natural
frequency, thus the harmonic motion of the system will change its critical frequency point
corresponded with the unphysical jump. For this parametric resonance response case, the Eq.
(26) in the chapter 3.1 is changed to
10
*
TT k   (38)
This slight change means the secular terms with okTi
e 
which will be eliminated in the
follow steps becoming,
0)3(
4
3
2
3
442
'2-
2
3311
22
33
22
33
23
33
2
11111
*
1
11
11




AAgAgeAAgeAAg
eAgeAgAggAibgiA
kkkk
Ti
kk
Ti
kk
Ti
kk
Ti
kkkkkkkkkk













(39)
Comparing Eqs. (27) and (39), one may find Eq. (27) presenting nonlinear asymmetric
harmonic resonance as well as Eq. (39) showing symmetric harmonic resonance. Similar to the
procedures in primary response, we replace i
aeA
2
1
 and denote,
  1T (40)
Solving the Eq. (39) at the steady-state condition which means:
0 a (41)
The frequency-amplitude and phase-amplitude differential equations are given as:

37. 17



2sin
168
2 3
31131111
111
* 








agag
gb
a
k
(42)
























16
3
2cos
8
2cos
8
48
3
5.0
1
3
3113
3
31131111
1111
3
3113
1111
111
ag
agag
agag
ag
ag k








 (43)
By expanding van de Waals term and electrostatic term with Taylor expansion in
numerator at 0 point (refer table 1), one input those coefficients with system constants in Table 2,
dimensionless parameters from table 4. As mentioned, primary resonance response, Fig.3, and
parametric resonance response, Fig. 4, will be investigated. Therefore, natural frequency 1
(when k=1) list in Table 3 should be used.
3.3. Frequency response for primary and parametric resonances using MMS
The constants, dimensional parameters, and dimensionless parameters, respectively, used
in this work are given in Tables 3-5. Table 4 listed the dimensional values of those parameters.
Table 3. System Constants [33, 24]
Description Symbol Value (unit)
Hamaker constant H 6.0e-19
J
Permitivity of free space ε0 8.854e-12
C2
/N/m2
Table 4. Dimensionless Parameters of the System [23]
Description Symbol Value (unit)
Thickness of nanotube h 0.35*10-9
m
Length of CNT l 200*10-9
m
CNT radius r 10-9
m
Gap CNT-CNT g 42*10-9
m
Young's modulus E 1.2*1012
Pa
Density of material ρ 1.33*103
kg/m3

38. 18
Voltage V0 0.067 V
Table 5 gives the dimensionless values for voltage, damping and van de Waals
parameters.
Table 5. Dimensionless System Parameters (calculated by Eqns 8, 9 and 10)
Description Symbol Value
Electrostatic constant δ 0.150
Damping constant b* 0.001
Van De Waals constant μ 0.005
With the data from Table 5, the frequency responses for primary and parametric resonances
are given in Figs. 2 and 3, respectively.
Figure 2. Amplitude frequency response of primary resonance using MMS. AC frequency near
half fundamental natural frequency 1=3.5165. = 0.005, = 0.15, b*=0.001.

39. 19
Figure 3. Amplitude frequency response of parametric resonance using MMS. AC frequency
near fundamental natural frequency 1=3.5165. = 0.005, = 0.15, b*=0.001.

40. 20
CHAPTER IV
REDUCED ORDER MODEL FOR ELECTROSTATIC ACTUATION
The Reduced Order Model (ROM) method will be used to numerically model this
electrostatic nanotube nanotweezers system. This method can accurately model and present the
behavior of the system without making assumptions that the nonlinearities of the system are weak.
High amplitude vibrations and low amplitude oscillations will be shown in figures In order to use
ROM, the Taylor expansion of Eq. (13) is given in the denominators as


*cos
11 2
5
0
5
0
4
4
*
2
2









 
k
k
k
k
k
k wbwa
z
ww
b
t
w
(44)
Table 6. Coefficients of Taylor expansions in the denominator.
Symbol Value Symbol Value
0a 27.2157 0b 1
1a -83.9424 1b -5
2a 37.5110 2b 7.5
3a 19.6740 3b -2.5
4a 17.0074 4b -0.625
5a 18.2755 5b -0.375
6a 22.2340 6b -0.3125
7a 29.3248 7b -0.3125
This will better approximate the singular points of the electrostatic and van der Waals
forces, leading to more realistic results in higher amplitudes. The ROM solution of Eq. (44) has
the form of

41. 21
     

N
j
ii zuzuw
1
,  (45)
Where N is the number of terms,  iu are time functions to found through integration and
 zi the mode shape of nanotubes. Multiplying Eq. (44) with both denominators in Eq. (44) and
the substituting Eq. (45) into the resulting equation, it results
     

5
0
2
5
0
)4(*
5
0
5
0, 1
1
1
2
2
2
1 2
2
2
1
1
*cos
k
k
k
k
k
k
k k
k
k
k
k uuuubuuu   (46)
Where jjj w  2)4(
 . Hence, we derived the follow equation,


*cos2
5
0 1
5
0 1
1
)4(
1
*
1
5
0,0 1
1
1
1
2
2
2
21
21
21










































  
 
  



k
k
N
j
jjk
k
k
N
j
jjk
N
j
jj
j
N
j
jj
N
j
jj
kk
kk
N
j
jjkk
uu
uubuu 
(47)
Or






*cos......
......
......
......
......
2
0
5
1
5
1...1,1
11
0
5
0,
5
1...1,1
11
00
1
2
1
*
1
5
1
5
1...1,1
110
1
2
1
*
1
5
1
5
1...1,1
110
1
2
1
*
1
5
1,1
5
1...1,1
11
1
2
1
*
1
2 11
221
2 21
222
2 21
222
1 11
111
21 211
212121























































 
 

 
 
 
 
 

 
 
  

k jjj
jjjjjjk
k jjj
jjjjjjk
j
N
j
jj
j
N
j
jj
N
j
jj
k jjj
jjjjjjk
N
j
jj
j
N
j
jj
N
j
jj
k jjj
jjjjjjk
N
j
jj
j
N
j
jj
N
j
jj
kk jjj
jjjjjjkk
N
j
jj
j
N
j
jj
N
j
jj
k
kk
k
kk
k
kk
k
kk
kk
kkkk
uuu
uuu
uubu
uuuuubu
uuuuubu
uuuuubu




(48)
Denote,

42. 22
j
k
N
jjj
jjjjjjk
k
N
jjj
jjjjjjk
kk
N
jjj
jjjjjjkkj
k
kk
k
kk
kk
kkkk
uuu
uuu
uuuD



00
5
1 1...1,1
110
5
1 1...1,1
110
5
1,1 1...1,1
11
2 21
222
1 11
111
21 211
212121
......
......
......



 
 
 
 
 
  

(49)
Substituting Eq. (49) back to Eq. (48),



*cos......
......
2
0
5
1 1...1,1
11
0
5
0 1...1,1
11
1
2
1
*
1
2 11
221
2 21
222






























 
 

 
 

k
N
jjj
jjjjjjk
k
N
jjj
jjjjjjk
j
N
j
jj
j
N
j
jj
N
j
jj
k
kk
k
kk
uuu
uuu
Duubu 
(50)
Then, multiplying Eq. (50) by
m , integrate from 0 to 1, it results



*cos...
...
2
0
3
1
3
1...1,1
...1
0
5
0 1...1,1
...1
1
2
1
*
1
2 11
1121
2 21
2122





















 
 

 
 

k jjj
jmjjjjjk
k
N
jjj
jmjjjjjk
N
j
mjjj
j
N
j
mjjmj
N
j
j
k
kk
k
kk
huuu
huuu
EuEubEu 
(51)
Where m = 1, 2, …, N and
mj
k
N
jjj
jmjjjjjk
k
N
jjj
jmjjjjjk
kk
N
jjj
jmjjjjjkkmj
hhuuu
huuu
huuuE
k
kk
k
kk
kk
kkkk
00
5
1 1...1,1
...10
5
1 1...1,1
...10
5
1,1 1...1,1
...1
2 21
2122
1 11
1111
21 211
2112121
...
...
...





 
 
 
 
 
  

(52)
And

43. 23
dzDE jjmj 
1
0
 (53)
dzh mm 
1
0
 (54)
dzh jmmj 
1
0
 (55)
dzh kk jjjmjmjj 111
..1
1
0
...  (56)
dzh kk jjjmjmjj 221
..1
1
0
...  (57)
dzh kkkk jjjmjmjj 21211
..1
1
0
...    (58)
When m = 1, denote 1B as



*cos...
...
2
0
5
1 1...1,1
...1
0
5
0 1...1,1
...1
1
1
2
1
1
*
1
1
1
1 11
1111
2 21
2122





















 
 

 
 

k
N
jjj
jmjjjjjk
k
N
jjj
jmjjjjjk
N
j
jjj
j
N
j
jjj
N
j
j
k
kk
k
kk
huuu
huuu
EuEubEuB 
(59)
Similarly, when m =2, denote 2B as

44. 24



*cos...
...
2
0
5
1 1...1,1
...1
0
5
0 1...1,1
...1
1
2
2
1
2
*
2
1
2
1 11
1111
2 21
2122





















 
 

 
 

k
N
jjj
jmjjjjjk
k
N
jjj
jmjjjjjk
N
j
jjj
j
N
j
jjj
N
j
j
k
kk
k
kk
huuu
huuu
EuEubEuB 
(60)
The system of N second order differential Eq. (59) is then transformed into a system of
2N first order differential equations. In the case of N = 2, denote,
24
23
12
11
uy
uy
uy
uy






(61)
One can easily derived that,
24
12
yy
yy




(62)
 ..2,1,122   nyy nn
 (63)
Finally, the system of first order differential equations in matrix form, can be written as





































2
4
1
2
4
3
2
1
2221
1211
00
0100
00
0001
B
y
B
y
y
y
y
y
EE
EE




(64)
Setting time span, the numerical simulation conducted using MATLAB give the time
response as in Fig. (4). One can notice that the steady-state is reached around 000,10 . An
initial amplitude of 2.00 U is considered. Umax is the amplitude of the tip of the nanotube.

45. 25
Figure 4. Three terms ROM time response.

46. 26
CHAPTER V
HALF NATURAL FREQUENCY RESONANCE RESPONSE
5.1 Amplitude-Frequency Response for Primary case.
Using the system parameters, and dimensional parameters, with the values shown in Tables
3-6, the nanotweezer’s amplitude-frequency response for an actuation of AC frequency near half
natural frequency of the nanotweezers, which leads to primary resonance, is shown in Fig. 5. Solid
lines show stable steady-state amplitudes, and dash lines unstable steady-state amplitudes. One
can see the amplitude increasing with the increase of AC frequency in a range of  = -0.04 to  =
0, where  = 0 corresponds to half natural frequency of the nanotweezers. These oscillation
amplitudes keep stable and increase until a saddle-node point B reached. At this point the
nanotweezers system suddenly become unstable and the oscillations amplitudes jumped to high
amplitude on the right hand side solid branch or pull-in. The saddle-node point bifurcation point
B is very important in MEMS and NEMS systems. After that point, the oscillation amplitude
corresponded to further detuning frequency goes higher than 0.5 of tweezers gap, which means
two carbon nanotweezers get in contact with each other, i.e a pull-in phenomenon occurs. In the
case of the resonator starting from rest, which means zero initial amplitude, the resonator
amplitude will increase settling to a stable steady-state amplitude on the solid line. Conversely, if
the resonator starts from an initial amplitude above the dash line, the amplitude increases up to 0.5
of the nanotweezers gap, which means the two tweezers will make contact, a pull-in phenomenon
occurring.

47. 27
Figure 5.Amplitude frequency response of primary resonance using MMS and 3T ROM. AC
frequency near half fundamental natural frequency 1=3.5165. = 0.005, = 0.15, b*=0.001.
The amplitude-frequency response, depicted as solid and dash line in Fig. 5, was verified
by those solid circles which are numerical time responses obtained from Matlab with 3 terms
ROM. One should notice that the pull-in phenomenon happens in a range of frequencies no matter
what the initial amplitude is. We call this escape band as we indicated in Fig. 5. Some specific
frequencies like detuning frequency,  = -0.03,  = -0.02,  = -0.01 were used to obtain time
response of 3T ROM. Their amplitude corresponded to steady-state were measured and their
values were indicated as solid circles in Fig 5. Then one reversed the detuning frequency from
high detuning frequency (= 0.04) back to its natural frequency, similar behavior were observed.
Resonance amplitude obtained from time response also showed consistent agreement with the
MMS frequency response, but MMS fails to predict the nonlinearity of nanotweezers higher
amplitude.

48. 28
Figure 6 illustrates the phase-frequency response of the nanotweezers system using
MMS. The AC frequency is near half natural frequency of nanotubes. The solid line at the top of
Fig. 6 shows the phase values of the solid stable branch in Fig. 5. Similar correspondence
between the lower branches, stable and unstable, in Fig.6 and the left-hand side branch in Fig 5,
occurs. The star at the end of low solid branch and the start of the dash branch shows the point
corresponding to the saddle bifurcation point B in Fig 5.
Figure 6. Phase frequency response of primary resonance using MMS. AC frequency near
half fundamental natural frequency 1=3.5165. = 0.005, = 0.15, b*=0.001,1=3.5165.
Figure 7 shows the phase plane response of the steady state points showed in Fig. 5. One
can notice that all of them are limit cycles. As the frequency increases, the amplitude increases.

49. 29
Figure 7. Phase plane for three steady-state points P1, P2, and P3 in Fig. 5. AC frequency
near half natural frequency 1=3.5165. = 0.005, = 0.15, b* = 0.001.
Figure 8 shows the three terms ROM time response of the nanotweezers. The detuning
frequency and the initial amplitude were chosen such that the initial point is above the left solid
branch in Fig. 5. This is used to test the existence of the left stable solutions using the time
response. As Fig. 8 shows, the initial amplitude U0 is given as U0 =0.1 of the gap of
nanotweezers and the detuning frequency keep a constant at  = -0.03.One can see that the
oscillation at the tip of the nanotweezers gradually decreases to an amplitude Umax= 0.03 where
the system reaches its steady state.

50. 30
Figure 8. Time response using 3T ROM. AC frequency near half fundamental natural
frequency 1=3.5165. U0 = 0.10, = 0.15, = 0.005, = -0.03, b* = 0.001.
Figure 9 shows a new particular point of  = -0.008. The initial amplitude for this detuning
frequency was U0 = 0.1 of the gap, which is lower than the unstable branch. The time response
results in a steady state amplitude Umax = 0.132 which is are located on lower stable branch.
Figure 5 and 9 are in agreement. One can refer point A in Figure 10, which is a zoom of
Fig. 5.

51. 31
Figure 9. Time response using 3T ROM. AC frequency near half fundamental natural
frequency 1=3.5165. U0 = 0.10, =0.15, = 0.005, = -0.008, b*=0.001.
Figure 10. Zoom in of Fig. 5. AC frequency near half fundamental natural frequency
1=3.5165. = 0.005, = 0.15, b*=0.001.

52. 32
Figure 11 shows that at the same detuning frequency but with an initial amplitude of 0.45
of gap, which is above the dash line in Fig. 5, or point B in Figure 10. The system goes into pull-
in, meaning an amplitude of 0.5 of the gap is reached. This is in agreement with Figs. 5 or figure
8.
Figure 11. Time response using 3T ROM. AC frequency near half fundamental natural
frequency 1=3.5165. U0 = 0.45, =0.15, = 0.005, = -0.008, b*=0.001.
Figure12 shows the time response of a CNT nanotweezer, which is actuated by an AC
voltage at a detuning frequency of  = -0.0055 with an initial amplitude value 0.2. This is an initial
point in the escape band. The response shows the oscillations amplitude at the free end of nanotube
increases from U0 = 0.2 until collapses reaching an amplitude at 0.5 of gap which located on the
right solid branches. This is in agreement with the frequency response given by the three terms
ROM shown in Fig. 5.

53. 33
Figure 12. Time response using 3T ROM. AC frequency near half fundamental natural frequency
1=3.5165. U0 = 0.2, = 0.005, = 0.15, b* = 0.001, = -0.0055.
Figure 13, shows the time response using the three terms ROM for a point of initial
coordinates in Fig. 5, U0 = 0.1 and = 0.03. The steady state amplitude given by the time response
is Umax = 0.02. This is in agreement with the right stable solutions shown in Fig. 5. Here one may
notice the limitation of MMS amplitude frequency response.

54. 34
Figure 13. Time response using 3T ROM. AC frequency near half fundamental natural frequency
1=3.5165. U0 = 0.1, = 0.005, = 0.15, b*=0.001, = 0.03.
As shown in Fig. 5, the amplitude-frequency response given by MMS in good agreement
with time-response given by three terms ROM at low amplitude resonance below 0.2 of the
nanotubes gap. But at the high amplitude resonance, MMS fails to provide accurate solutions.
Figure 14 shows the effect of voltage on the amplitude-frequency response. Three cases
were investigated and compared  = 0.1,  = 0.15, and  = 0.2. The other parameters, Van de
Waals and damping, were held constant,  = 0.005, b*=0.001. With the increase of the excitation
parameter , the steady state vibration amplitude increases; the bifurcation points which are
depicted by stars and its corresponded frequency increase too, only the frequency corresponded
to unstable branches decrease due to pull-in phenomenon occurs needs large oscillation
amplitudes, and this can be the explanation of escape band becoming larger. The softening effect
increase with the increase of voltage parameter.

55. 35
Figure 14. Effect of voltage parameter  on the amplitude frequency response using 3T ROM.
AC frequency near half fundamental natural frequency 1=3.5165. = 0.005, b* = 0.001.
Figure 15 depicts the effect of the damping parameter b* to the amplitude-frequency
response of the nanotweezers’ system. By increasing the damping parameter input to the system
from b*=0.001 to b*=0.008, the amplitude at steady-state of the resonator decreases, as well as
the escape band; as the damping parameter goes up to b* = 0.008, the pull-in phenomenon no
longer occurs and the softening effect as well as escape band are not significant. If one system
has high damping, then the pull-in phenomenon does not occur anymore. The bifurcation
frequency shifts slightly to higher frequency as the damping increases.

56. 36
Figure 15. Effect of damping parameter b* on the amplitude frequency response using 3T ROM.
AC frequency near half fundamental natural frequency 1=3.5165. = 0.005, =0.15.
Figure 16 shows the effect of van de Waals parameter to the amplitude-frequency
response of the nanotweezer system. In order to investigate this effect, we varied  from =0.0005
to =0.005. From Fig. 16, one can clearly observe the softening effect increase with the increase
of . Conversely, the escape band reduces. This means that van de Waals parameter affects
significantly the frequency range the system experiences pull-in regardless the value of initial
amplitude. The bifurcation frequency is shifted lo lower as the van de Waals parameter increases.
From Figs. 14, 15, and 16, one can conclude that MMS is in agreement with ROM when
the oscillation of steady state amplitude is less than 0.3 of the gap. However, MMS fail to predict
accurately the behavior of the system for larger amplitudes.

57. 37
Figure 16. Effect of van der Waals parameter  on the amplitude frequency response using 3T
ROM. AC frequency near half fundamental natural frequency 1=3.5165. b*= 0.001, =0.15.
Figure 17 show the convergence of the reduced order model, and it is composed of MMS
(can be regard as one term reduced order model), two term ROM as well as three terms ROM.
One can observe those three being in agreement in low amplitudes. The tendency of these
branches given a good illustration of the convergence of the ROM time response method. The
simulations shows that 3 terms ROM better predict the pull-in phenomenon, and the behavior in
high amplitudes. The bifurcation point is predicted the same by all three methods. Therefore any
of these methods can be used to predict the behavior of the system before pull-in phenomenon
occurs.

58. 38
Figure 17. Convergence of the amplitude frequency response showing MMS, two terms (2T)
ROM, and three terms (3T) ROM. AC frequency near half fundamental natural frequency
1=3.5165. = 0.005, = 0.15, b* = 0.001.
5.2. Voltage-Amplitude Response for Primary Resonance.
Figure 18 compares MMS and 3 terms ROM voltage-amplitude response. A two point
bifurcation diagram was found in this case. This figure shows relationship between voltage and
amplitude for the investigated system. A certain range of AC voltage has been actuating the
system while keeping the detuning frequency a constant parameter.
The steady-state solution of voltage response shows one branch with two saddle-node
bifurcation A and B, which led the system to a phenomenon of hysteresis. The dash vertical lines
shows the hysteresis loop boundaries. As depicted by arrows, solutions between A and B are
unstable solution which are represented by the dash lines, any points located there would either
go up above solid solutions or go down to stable solutions on lower solid line. This oscillation
started from rest.

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Figure 18. Amplitude voltage response of primary resonance using MMS and 3T ROM. AC
frequency near half fundamental natural frequency 1=3.5165. = 0.005, = -0.004, b*=0.001.
The nanotweezers are actuated by AC voltage of frequency = -0.004. With the increase
of voltage, the steady-state amplitude increases along the stable branch until reaches the
bifurcation point A. The system then loses its stability and jumps to higher amplitudes. In the
case of non-zero initial amplitudes below the dash branch, the amplitude decreases until it
reaches the stable (solid line) lower branch.
Conversely, in the other case of initial amplitude above the unstable branch (dash line)
the amplitude increases to settling on the upper solid branch (see 3T ROM in Fig. 18). For
amplitude lower than 0.3 of the nanotubes gap, the MMS were shown in agreement with 3 terms
ROM, which can be observed from Fig. 18. As a remainder, MMS shows an accurate and
reliable predict of weak nonlinearities electrostatically system with at lower amplitude.

60. 40
Figure 19 shows the phase plane response of the steady state points showed in Fig. 18.
One can notice that all of them are limit cycles. As the frequency increases, the amplitude
increases.
Figure 19. Phase plane for three steady-state points P4, P5, and P6 in Fig. 19. AC frequency
near half natural frequency 1=3.5165. = 0.005, = 0.15, b* = 0.001.
Figures 20 - 24 are time responses given by numerical simulations with 3 term ROM.
They are in agreement with Fig. 18. Figure 20 shows the time response for an initial amplitude
U0 = 0.1, and AC voltage = 0.01 and detuning frequency = -0.004. The amplitude of the tip
of the nanotube decreases and settles to Umax = 0.018 reaching steady state. This in agreement
with the low solid branch given by three terms ROM frequency response in Fig. 18.

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Figure 20. Time response using 3T ROM. AC frequency near half fundamental natural frequency
1=3.5165. U0 = 0.1, =0.01, = 0.005, = -0.004, b*=0.001.
Figures 21 and 22 illustrate the time responses for initial amplitude U0 = 0.2 and = 0.04,
U0 = 0.3 and = 0.04. Figure 21 shows that the amplitude at the tip decreased from initial
amplitude U0 = 0.2 to its steady state amplitude Umax= 0.095. Given the end of nanotube an
initial amplitude U0 = 0.2 of the nanotweezer gap and voltage parameter  = 0.04, this means an
initial point below the unstable dash branch, but above the lower stable solid branch. The results
given by time response the steady state amplitude was 0.085 in Fig. 21, which is in agreement
with Fig. 18.
In Fig. 22 for the same voltage = 0.04, but an initial amplitude above the dash branch,
the amplitude of the tip increased to 0.378 of the gap. This is in agreement with the fact that the
dash line represents saddle points, which are unstable, and therefore the amplitude moves away
from the unstable saddle steady-states.

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Figure 21. Time response using 3T ROM. AC frequency near half fundamental natural frequency
1=3.5165. U0 = 0.20, =0.04, = 0.005, = -0.004, b*=0.001.
Figure 22. Time response using 3T ROM. AC frequency near half fundamental natural frequency
1=3.5165. U0 =0.30, =0.04, = 0.005, = -0.004, b*=0.001.

63. 43
Figure 23 shows a case of an initial amplitude lower than the lower branch, with a
voltage of  = 0.05 and an initial amplitude of U0 = 0.1. The amplitude settles to a steady-state
amplitude on the lower stable branch. Its amplitude as shown in Fig. 20 is 0.125, located on the
low solid branch, and in agreement with Fig. 18.
Figure 23. Time response using 3T ROM. AC frequency near half fundamental natural frequency
1=3.5165. U0 = 0.1, =0.05, = 0.005, = -0.004, b*=0.001.
Figure 24 shows a case when the amplitude would settle to the higher solid branch. When
the voltage is  = 0.07 and the initial amplitude U0 = 0.1, the amplitude of the nanotube tip
becomes Umax = 0.378 of the gap, which located on the higher solid branch. This is in agreement
with Fig. 18.
Figure 25 illustrates the influence of detuning frequency on the voltage response while
AC near its half natural frequency 1=3.5165.

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Figure 24. Time response using 3T ROM. AC frequency near half fundamental natural frequency
1=3.5165. U0 = 0.1, =0.07, = 0.005, = -0.004, b*=0.001.
Figure 25. Effect of detuning frequency parameter on the amplitude frequency response using
3T ROM. AC frequency near half fundamental natural frequency 1=3.5165.=0.005,b*=0.001.

65. 45
Three cases = -0.006, = -0.003, and -= 0.001 were investigated using MMS and 3T
ROM. As one can observe that the amplitude of the nanotweezers gradually changes from a
strong nonlinear behaviors to a linear behavior as the detuning frequency increases from = -
0.006 up to = -0.001. After the detuning parameter goes up to = -0.001, the oscillation of
nanotube tweezers are stable and no longer experience a bifurcation or pull-in phenomena. The
nanotweezers system found its maximum amplitude Umax = 0.35, Umax = 0.27 for detuning =-
0.003 and for detuning =-0.001, respectively. One may observe that MMS and ROM are in
agreement for amplitudes below 0.2 of the gap. For larger amplitudes only 3 terms ROM
provides accurate results.
Figure 26 shows the influence of damping parameter  on the voltage-amplitude
response, with holding the detuning frequency at = -0.004. The voltage response curves of the
nanotubes gradually shift from nonlinear behavior to linear behavior with the increase of
dimensionless damping coefficients from b*=0.001 to b*=0.008. For large damping of b*= 0.006
and higher, the system does not experience bifurcation or pull-in phenomena. In all three cases,
the voltage response using MMS and ROM are in agreement for amplitudes lower than 0.2 of the
gap.
Figure 27 shows the influence of dimensionless van de Waals parameter on the voltage
response, while keep detuning frequency  = -0.005 and damping parameter b*=0.001. Different
than frequency and damping parameters, the increase of van de Waals parameter would not
lead the system from nonlinear to linear behavior. Increasing van de Waals parameter from
=0.0005 to =0.005, the nonlinear effect increases, and the bifurcation voltage decreases,
which means that the system goes into a pull-in or jump to higher branch phenomena. MMS is in
agreement with ROM for amplitudes lower than 0.2. MMS gives reliable solutions at lower

66. 46
amplitudes while for higher amplitudes and jump and pull-phenomena one should use three term
ROM.
Figure 26. Effect of damping parameter b*on the amplitude frequency response using 3T ROM.
AC frequency near half fundamental natural frequency 1=3.5165. = 0.005, b*=0.001.
Figure 27. Effect of damping parameter b*on the amplitude frequency response using 3T ROM.
AC frequency near half fundamental natural frequency 1=3.5165.  = -0.004, b*=0.001.

67. 47
CHAPTER VI
NATURAL FREQUENCY RESONANCE RESPONSE
6.1 Amplitude-Frequency Response for Parametric resonance.
Figure 28 gives the amplitude-frequency response for nanotweezers system when the AC
frequency is near natural frequency of the nanotubes. Two Hopf bifurcation points were found,
denoted as A and B. They are subcritical bifurcation point and supercritical bifurcation point,
respectively. One can observed from Fig. 28, the subcritical bifurcation point located at the
intersection of dash line and the horizontal  axis as well as supercritical bifurcation point
located at the intersection of solid line and horizontal  axis. The dash line represent the
unstable solutions and solid line represent stable solutions. One should mention that the
frequency response shows a softening effect (bending to the left of both branches).
With the frequency swept up, increasing the detuning frequency, the nanotubes keep zero
steady-state amplitude. Zero amplitude is part of the MMS solution of the system. When the
detuning frequency reaches the frequency of the subcritical A bifurcation point, they become
unstable, so the free end of the nanotubes instantly jumps to an amplitude of the solid branch
above. As the detuning frequency continues to increase, the amplitude gradually decreases until
the zero amplitude of the supercritical bifurcation point B, after which the right end of nanotubes
keeps zero amplitude steady-state.

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Figure 28. Amplitude frequency response of primary resonance using MMS and 3T ROM. AC
frequency near fundamental natural frequency 1=3.5165. = 0.005, = 0.15, b*=0.001.
Figure 29 shows the phase-frequency response with AC frequency near natural frequency
of the nanotubes. The solid line at the top corresponds to the solid MMS stable branch in Fig. 28
and the dash branch corresponds to the unstable MMS dash branch in Fig. 28.
Since MMS fail to predict higher amplitude and strong nonlinearities, one can only
accurately predict the bifurcation points A and B, and amplitudes below 0.2 of the gap.
However, MMS has its advantages. It rapidly and accurately present the frequency response in
low amplitudes of a weakly nonlinear system. Another factor affecting the oscillation of the
nanotubes are the initial amplitudes. In the case of an initial amplitude below the unstable branch
for a given frequency , the amplitude the free end decreases to zero. Otherwise, in the case an
initial amplitude above the dash line, the amplitude of the free end of the nanotube increases to
reach the solid line above.

69. 49
Figure 29. Phase frequency response of primary resonance using MMS. AC frequency near
fundamental natural frequency 1=3.5165. = 0.005, = 0.15, b*=0.001.
To test the correction of frequency response obtained from three terms ROM, one uses
time responses, as before. Time response were shown in Fig. 30 with dimensionless parameter
values  = 0.15, b*=0.001,  = 0.005,  = -0.0125 and initial amplitude of U0 = 0.35, which is
located below the solid branch but higher than the dash branch in Fig. 28. The steady-state
amplitude at this frequency, Fig. 28, is 0.40 of the gap. As one can observe the time response
results given in Fig. 30 is in agreement with Fig. 28.
Figure 31 shows the time response for a point U0 = 0.025, = -0.01 located lower than
the dash branch and at the left hand side of point A. The results show that the oscillation of the
nanotubes tip decrease from the initial amplitude U0 = 0.025 to zero amplitude, in agreement
with the frequency response given by Fig. 28.

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Figure 30. Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0.35. =0.15, = 0.005, = -0.0125.
Figure 31. Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0.05. =0.15, = 0.005, = -0.015.

71. 51
Since zero amplitude is always a steady-state solution of the system, one may test the
behavior of the system between subcritical bifurcation point A and supercritical bifurcation points
B. Figure 32 shows the time response for a point of frequency  =- 0.006 and initial amplitude of
U0 = 0, Fig. 25. One can see that the amplitude at the free end of the nanotubes increases with time
and settles to an amplitude Umax = 0.24. This is in agreement with Fig. 28.
Figure 33 shows the time response for an initial point located above the solid branch and
at the left hand side of the supercritical bifurcation point B. This is a point of initial amplitude of
U0 = 0.4 and detuning frequency  = - 0.01. The amplitude of the tip of the nanotubes decreases
and settles to an amplitude of Umax = 0.36 of the gap, then the nanotweezers system kept stable.
This is in agreement with Fig. 28.
Figure 32. Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0, =0.15, = 0.005, = -0.006.

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Figure 33. Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0.4, =0.15, = 0.005, = -0.01.
Figure 34 completes the testing of the frequency response given by three terms ROM and
MMS in Fig. 28. The time response in Fig. 30 shows the existence of zero solid branch at the
right hand of the supercritical bifurcation B point. The initial amplitude is U0 = 0.02, and the
detuning frequency  = -0.003. The amplitude at the tip of nanotubes decreases and reaches
steady state with a zero amplitude. This is in agreement with Fig. 25.

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Figure 34. Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0.1, =0.15, = 0.005, = -0.003.
In order to investigate the influence of the voltage on the frequency response, three
values of the dimensionless parameter  are considered in Fig. 35,  = 0.1,  = 0.15, and  = 0.2
using both MMS and three terms ROM. With the increase of AC voltage, the softening effect
becomes stronger. The range of frequency between subcritical bifurcation point and supercritical
bifurcation point increases, and both bifurcation points move to lower frequencies. The solid
branches are shifted less than the dash branches as the voltage increases. MMS fails to predict
the behavior for high amplitudes and strong nonlinearities. MMS does not show the correct
behavior of nanotubes for amplitude higher than 0.2 gap of the nanotweezers. However, there is
an agreement between MMS and ROM for amplitudes less than 0.2 of the gap.

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Figure 35. Effect of voltage parameter  on the amplitude frequency response using 3T ROM.
AC frequency near fundamental natural frequency 1=3.5165. = 0.005, b*=0.001.
Figure 36 gives the influence of dimensionless damping b*
on the amplitude response,
using both MMS and 3 terms ROM. Three cases b*
=0.001, b*
=0.002 and b*
=0.008 are
investigated. The other parameters are the same = 0.005, =0.15. As the damping increases,
the softening effect of the solution reduces, and the range between of the subcritical bifurcation
point and supercritical bifurcation point becomes smaller.
Figure 37 shows the van de Waals force effect on the frequency response. Other
parameters like voltage , damping b*
are b*
= 0.001 and =0.15, while three dimensionless van
de Waals values are = 0.004, = 0.005 and = 0.006. Both MMS and 3 terms ROM are used.

75. 55
Figure 36. Effect of damping parameter b*on the amplitude frequency response using 3T
ROM. AC frequency near fundamental natural frequency 1=3.5165. = 0.005, =0.15.
Figure 37. Effect of damping parameter on the amplitude frequency response using 3T ROM.
AC frequency near fundamental natural frequency 1=3.5165. b*
= 0.001, =0.15.

76. 56
Figure 38 shows the convergence from MMS (one term reduced order model) to three
term ROM frequency response. Comparing MMS, two terms ROM and three terms ROM, the
conclusion is that three terms ROM is needed to investigate the system; it captures better the
softening effect. The end points of the dash branches and solid branches move left with the
increase of number of terms. MMS and ROM agree for amplitudes less than 0.2 of the gap.
MMS fails to predict the nonlinear behavior in high amplitudes. Three term ROM gives more
reliable solutions than two terms.
Figure 38. Convergence of the amplitude frequency response showing MMS, two terms (2T)
ROM, and three terms (3T) ROM. AC frequency near fundamental natural frequency 1=3.5165.
= 0.005, = 0.15, b*=0.001.
One may observe that with the increase of van de Waals parameter, the bifurcation points
are shifted to lower frequencies, so the system loses stability for a lower frequency The range of

77. 57
frequencies between bifurcations remains the same regardless van de Waals parameter. The
softening effect seems not to be affected by van de Waals forces.
6.2. Voltage-Amplitude Response for Parametric Resonance.
The voltage-amplitude response of a CNT based nanotweezers under AC electrostatic
actuation of AC frequency near natural frequency is investigated next. MMS, and three terms
ROM results are shown in Fig. 39. The voltage-amplitude response shows three bifurcation
points, marked as A, B, C respectively.
These three bifurcation points are the critical points where the stability of the system
changes; one may observed from Figure 39, points A and C are Hopf bifurcation points, while A
is called subcritical bifurcation point and C is called supercritical bifurcation point. Point B is a
saddle-node bifurcation point and is located at the top end of dash branch where it merges with
the solid branch. The dash branch denotes unstable steady-state amplitudes while the solid
branch denotes the stable steady-state amplitudes. In Fig. 39, as the voltage swept up from the
dimensionless value =0 to the value =0.3, the amplitude at the free end of the nanotubes
remains zero amplitude until the voltage of the subcritical bifurcation point A is reached. At
Point A the system loses stability and the amplitude suddenly jumps from zero to a point of about
0.28 of the gap, point locate on the solid branch above. As the voltage continues to increase, the
resonator amplitude gradually decreases to zero amplitude along branch BC until it reaches the
supercritical bifurcation point C. From this point further the amplitude remains zero regardless
the increase in voltage. Backwards, if the voltage swept down, the amplitude starts to increase
from zero amplitude along branch BC until it reaches bifurcation point B, where it loses stability
and jumps down to zero amplitude. Continuing to decrease of the input voltage would not change

78. 58
the zero amplitude. MMS fails to accurately predict the behavior of the system for higher
amplitudes. One can observe that MMS and ROM are in agreement for amplitudes less than 0.2
of the gap. Three terms ROM better captures the softening effect.
Figure 39. Amplitude voltage response of primary resonance using MMS and 3T ROM. AC
frequency near fundamental natural frequency 1=3.5165. = 0.005, = -0.006, b*
=0.001.
To test the results shown in Fig. 39, time responses are simulated using three term ROM.
Figure 40 shows the time response corresponding to voltage = 0.05, frequency = -0.006, and
initial amplitude U0 = 0.2. This point is at the left hand side of the bifurcation point B. The
amplitude at the free end of the nanotubes decays to zero. This is in agreement with Fig. 39 since
the initial point is located below the unstable branch.
Figure 41 shows the time response in the case of voltage = 0.12, frequency = -0.006,
and initial amplitude U0 = 0.1. The amplitude at the free end of the nanotubes increase to a steady
state amplitude Umax = 0.27. This is in agreement with Fig. 39, the nanotubes from an initial point

79. 59
(amplitude) located below the stable branch BC and above the unstable AC, settle to an
amplitude on branch BC.
Figure 40. Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0.2, = 0.05, = 0.005, = -0.006, b* = 0.001.
Figure 41. Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0.1. =0.12, = 0.005, = -0.006, b*=0.001.

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Figure 42 illustrates the time response corresponding to an initial amplitude U0 = 0.1,
voltage = 0.08, and frequency = -0.006. The amplitude the tip of nanotubes decreases to zero
steady state amplitude. Since this point located below the unstable branch AB, Figs. 42 and 39
are in agreement, the system settles to a stable zero amplitude. One can see that from an initial
point close to the unstable branch AB, the system goes up to an amplitude on the stable branch
BC or to zero amplitude depending where the initial point is locate, below or above unstable
branch AB.
Figure 42. Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0.1. =0.08, = 0.005, = -0.006, b*=0.001
Figure 43 shows the time response for an initial point between the bifurcation points A
and C, and above the stable branch BC. Figures 43 and 39 are in agreement, both figures show
that the steady state amplitude in this case is Umax = 0.21, located on BC.

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Figure 43. Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0.30, =0.15, = 0.005, = -0.006, b*=0.001.
Figure 44 illustrates the time response using three terms ROM from an initial point at the
right hand side of the supercritical point C. The voltage is = 0.32, frequency = -0.006, and
initial amplitude U0 = 0.3. Both Figs. 39 and 44 predict a zero steady state amplitude. They are in
agreement.
Figure 45 shows the effect of frequency on the voltage response. Three dimensionless
discrete detuning frequency parameter values are chosen to investigate this effect, namely  = -
0.006,  = -0.0065 and  = -0.007. One can observe the shifting to lower voltage values of the
Hopf bifurcation points A and C as the detuning frequency increases. The range between
subcritical bifurcation point and supercritical bifurcation point becomes smaller with the increase
of the detuning frequency.

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Figure 44. Time response using 3T ROM. AC frequency near fundamental natural frequency
1=3.5165. U0 = 0.3. =0.32, = 0.005, = -0.006, b*=0.001.
Figure 45. Effect of detuning frequency parameter on the amplitude frequency response using
3T ROM. AC frequency near fundamental natural frequency 1=3.5165. = 0.005, b*=0.001.

83. 63
The oscillation amplitude at the free end of nanotubes increases with the decrease of the
detuning frequency. From this perspective, one can see that for = 0.15, and the three
frequencies considered (see the points in Figs. 45 and 46), the voltage response in Fig. 46 and
frequency response in Fig. 46 (a zoom in of Fig. 35) are in agreement. This can be regard as
example that the frequency response and voltage response are in agreement.
Figure 46. Zoom in of Fig. 35 for amplitude frequency response showing MMS, two terms (2T)
ROM, and three terms (3T) ROM. AC frequency near fundamental natural frequency 1=3.5165.
= 0.005, = 0.15, b*=0.001.
Figure 47 presents the influence of the damping parameter on the voltage response of
nanotubes. Judging from the voltage response given by MMS and three terms ROM, the increase
of the damping decreases the peak amplitude. Also, the range between subcritical point A and
supercritical point C becomes smaller, i.e. smaller range of voltage would lead the system into
nonzero amplitudes. The softening effect is not significantly influenced by damping.

84. 64
Figure 47. Effect of detuning frequency parameter b*on the amplitude frequency response using
3T ROM. AC frequency near fundamental natural frequency 1=3.5165. = 0.005, b*=0.001.
Figure 48 shows the influence of van de Waal parameter on the voltage response. As the
van de Waals parameter increases, the peak amplitude decreases, the amplitude of the saddle-
node bifurcation point B decreases, and both bifurcation points A and C are shifted to lower
voltage values. Also the range of voltages between A and C reduces, i.e. there is a smaller
interval of voltage values to reach non zero amplitudes. The voltage response given by MMS in
agreement with the time response using ROM in lower amplitudes.
Figure 49 shows the convergence of voltage response of nanotweezers using MMS (one
term ROM), two terms ROM, and three terms ROM when the AC frequency is near natural
frequency. One can notice that as the number of terms increases the behavior of the system is
better captured. Using three term ROM is better than using MMS. ROM better predicts the
behavior of nonlinear, even strongly nonlinear, systems and with large amplitudes. MMS is only

85. 65
valid for low amplitude and weak nonlinearities. However, one may see that MMS voltage
response, two term ROM and three ROM in agreement for lower amplitudes.
Figure 48. Effect of detuning frequency parameter on the amplitude frequency response using
3T ROM. AC frequency near fundamental natural frequency 1=3.5165.= -0.006, b*=0.001.
Figure 49. Convergence of the amplitude voltage response showing MMS, two terms (2T) ROM,
and three terms (3T) ROM. AC frequency near fundamental natural frequency 1=3.5165.=
0.005, = 0.15, b*=0.001.

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CHAPTER VII
DISCUSSION AND CONCLUSIONS
7.1. Summary
Two different methods were used to investigate the behavior of CNT nanotweezers
system. One method was an analytical, perturbation method, namely the Method of Multiple
Scales (MMS). The other method was the Reduced Order Model (ROM) method, with two and
three terms, for which a numerical integration is used. MMS and ROM results were compared.
The nanotweezers were actuated by an electrostatic force generated by a soft AC voltage.
The AC frequency considered in this thesis was near half natural frequency of the nanotweezer
leading the system into primary resonance, and near natural frequency of the nanotweezer
leading the system into parametric resonance. Three factors were taken into consideration to
effect the system responses, namely electrostatic actuation forces, van de Waals forces, and
damping forces. These three factor were present both in the frequency-amplitude response and
the voltage-amplitude response. In the frequency-amplitude response the frequency was the
independent variable and the voltage just a parameter, while in the voltage-amplitude response
the voltage was the independent variable and the frequency just a parameter.
MMS and ROM were in agreement for amplitudes less than 0.2 of the gap for all
responses of the CNT nanotweezers system. However, three terms ROM gave more accurate
results for amplitudes larger than 0.2 of the gap. One should mention that amplitudes of 0.5 of
the gap is a pull-in phenomenon. MMS had its disadvantages. It could not predict accurately the

87. 67
bifurcation points in moderate to high amplitudes. MMS is limited to small amplitudes and weak
nonlinear systems. However, it still a good method for rapidly generate accurate and reliable
results at low amplitudes and for weakly nonlinear systems. This drawback can be corrected in
higher amplitudes by using numerical integration of three term ROM, which gives accurate
predictions of moderate to high amplitude bifurcation points and the pull-in phenomenon of the
system. A convergence investigation based on the number of terms of the ROM was conducted,
and it was shown that the larger the number of terms in the ROM, the more accurate the results
are obtained.
Moreover, the frequency- amplitude response and voltage-amplitude responses given by
ROM for both primary resonance and parametric response were tested using time responses. All
time responses were in agreement with three terms ROM predictions.
The results for the nanotweezers system in terms of ROM time responses, frequency-
amplitude response, and voltage-amplitude response given by MMS and ROM were similar with
results of M/NEMS clamped circular plates [27], and literature for CNT resonators [23].
Previous research reported nanotweezers investigation using a lumped parameter model [32].
Also experimental data [33] has been reported. Both investigations provide similar behavior with
the results given in this work. As shown in Fig. 50, for a static simulation using the data of this
thesis, the distance between the tips of the nanotweezers reduces with the increase of the
actuating DC voltage from zero to 8 V. This in agreement with the tendency provide by
experimental data [33]. Recent work on MEMS and NEMS systems under other forces such as
Casimir was reported in the literature [34].

88. 68
Figure 50. Distance between nanotweezers with DC voltage. Pull-in voltage.= 0.005,
b*=0.001.
Boggild et al. [35] demonstrate a customization of nanotweezers with a 25nm gap
between arms. Nanotweezers with the gap of 42 nm of the nanotweezers system as in this work
can be used for manipulation of nanowires and floating nanostructures. Potential applications
include mounting components for nanosize devices, build electronic circuitry at nano level as
well as measurements.
7.2. Future Work
Further development of nanotweezers system investigation will include but not limited to
the behavior of nanotweezers for higher frequencies excitation. More phenomena like

89. 69
thermostatic damping will be taken into consideration. To test the analytical and numerical
predictions experimental work will be conducted.

90. 70
REFERENCES
[1] Iijima, S., 1991, “Helical microtubules of graphitic carbon”, Nature, Vol. 354, No. 6348, pp.
56–58.
[2] Binning, G., Rohrer, H., 1991, “Scanning Tunneling Microscopy-From Birth to Adolescece”,
Rev. of Mod. Phys, Vol. 59, No.3, pp. 615.
[3] Binning, G., Quate, C.F., Geber, Ch., 1986, “Atomic Force Microscope”, Phys. Rev. Letters,
Vol.56, No. 9, pp. 930.
[4] Dai, H., Hafner, J.H., Rinzler, A.G., Colbert, D.T., Smalley, R.E., 1996, “Nanotubes as
nanoprobes in scanning probe microscopy”, Nature, Vol. 384, pp.147-150.
[5] Nishijima, H., Kamo, S., Akita, S. Nakayama, Y., Hohmura, K.I., Yoshimura, S. H.,
Takeyasu, K., 1999, “Novel process for fabricating nanodevices consisting of carbon
nanotubes”, Appl. Phys., Vol.38, No. 7247, pp. 42 – 43.
[6] Eigler, D.M., Schweizer, E.K., 1990, “Positioning single atoms with a scanning tunneling
microscope”, Nature, Vol. 344, pp. 524.
[7] LI, Z.J., CHEN, X.L., DAI, L., 2002, “GaN nanotweezers”, Appl. Phys. Vol.76, pp.115-
118.
[8] P. Boggild, T. M. Hansen, K. Molhave, A. Hyldgard, M. O. Jensen, J. Richter, L. Montelius,
F. Grey, 2001,“Customizable nantweezers for manipulation of free-standing nanostructures
“, Conference paper.. DOL:10.1109/NANO.2001.966399. pp. 87 – 92.
[9] Philip Kim and Charles M. Lieber, 1999, “Nanotube Nanotweezer”, Report, Sicence,
Vol.286, No. 5447, pp. 2148-2150.
[10] Akita, S., Nakayama, Y., Mizooka, S., Takano, Y., Okawa, T., Miyatake, Y., Ymanaka, S.,
suji, M. T., Nosaka, T., 2001, “SPM Application of Carbon Nanotubes: Probes and
Tweezers”, Appl. Phys., DOI: 10.1109/IMNC. 984068, pp. 60 – 61.
[11] Lee J and Kim S., 2005, “Manufacture of a nanotweezer using a length controlled CNT
arm”. Sensors Actuators. doi:10.1016/j.sna.2004.11.012, Vol. 120, pp. 193-198.

91. 71
[12] Jiyoung, C., Byung-Kwon M., Jongbaeg K., Sang-Jo L. and Liwei L.,
2009,”Electrostaticallu actuated carbon nanowire nanotweezers, Smart Material and
Structures”. Vol.18, No. 065017, pp.7.
[13] Abdel-Rahman, E., Nayfeh, A., Younis, M., 2003, “Dynamics of an Electrically Actuated
Resonant Microsensor”, International Conference on MEMS, NANO and Smart Systems
(ICMENS’03), DOI: 10.1109/ICMENS.2003.1221991, pp.188-196.
[14] Miandoab, E. M., Pishkenari, H. N., Yousefi-Koma, A. ., 2014, “Chaos prediction in
MEMS-NEMS resonators”, International Journal of Engineering Science, Vol. 82, pp. 74-83.
[15] NikkhahBahrami, M., Ataei, A., 2010, “A Large Deflection Model for the Dynamic Pull-In
Analysis of Electrostatically Actuated Nanobeams in Presence of Intermolecular Surface
Forces”, Advanced research in Physics and Engineering, ISSN: 1790-5117, pp. 208-216.
[16] Caruntu, D., Martinez, I., Taylor, K., 2013, “Reduced order model analysis of frequency
response of alternating current near half natural frequency electrostatically actuated MEMS
cantilevers”, Journal of Computational and Nonlinear Dynamics, Vol. 8, pp. 031011-1 -
031011-6.
[17] Sasaki, N., Toyoda, A., Sayito, H., Itamura, N., Ohyama, M., Miura, K., 2006,
“Classification of Light-Induced Desorption of Alkali Atoms in Glass Cells Used in Atomic
Physics Experiments”, Journal of Surface Science and Nanotechnology 4,
DOI: 10.1380/ejssnt.2006, Vol.4, pp. 63-68.
[18] W. H. Lin, Y. P. Zhao. 2005, “Casimir effect on the pull-in parameters of nanometer
swiches”. Microsystem Technologies, Vol. 11, pp. 80-85.
[19] Koochi, A. , Fazli, N. Rach, R., Abadyan, M., 2014, “Modeling the pull-in instability of the
CNT-based probe/actuator under the Coulomb force and the van der Waals attraction”, Latin
American Journal of Solids and Structures, Vol.11, No. 8, pp. 1315-1328.
[20] Jiang, H., Hwang, K.C., Huang, Y., 2007, “Mechanics of Carbon Nanotubes: A Continuum
Theory Based on Interatomic Potentials”, Key Engineering Materials, Vol. 340-341, pp. 11-
20.
[21] Gholami, R., Ansari, R., Rouhi, H., 2015, “Studying the effects of small scale and Casimir
force on the non-linear pull-in instability and vibrations of FGM microswitches under
electrostatic actuation”, International Journal of Non-Linear Mechanics, pp. 193–207. Vol.
77, pp. 193–207.
[22] Wang, G. W., Zhang, Y., Zhao, Y. P., Yang, G. T. 2004, “ Pull-in instability study of
carbon nanotube tweezers under the influence of van der Waals forces”, Journal of
Micromechanics and Micro engineering Vol. 14, No. 8, pp.1119-1125.

92. 72
[23] Caruntu, D.I., Luo, L., 2014, “Frequency response of primary resonance of electrostatically
actuated CNT cantilevers”, Nonlinear Dynamics, Vol.78, pp.1827-1837.
[24] Caruntu, D.I., Martinez, I., 2014, “Reduced order model of parametric resonance of
electrostatically actuated MEMS cantilever resonators”, International Journal of Non-
Linear Mechanics, Vol. 66, pp. 28-32.
[25] Caruntu, D.I., Knecht, M., 2015, “MEMS cantilever resonators under soft AC voltage of
frequency near natural frequency”, Journal of Dynamic Systems, Measurement and Control,
Vol. 137, No.041016-1, DOI: 10.1115/1.4028887.
[26] Chen, X. Q and Saito, T., 2001, “Aligning single-wall carbon nanotubes with an
alternating-current electric field, American Institute of Physics, DOI: 10.1063/1.1377627.
[27] Caruntu, D. Oyervides, R, 2016, “Primary resonance voltage response of electrostatic
actuated M/NEMS circular plate resonators” DSCC2014-6277.
[28] Banks, T., “Modern Quantum Field Theory”, ISBN-13 978-0-511-42899-9.
[29] Bhiladvala, R. B. and Jane, Z. W., 2003, “Effect of fluids on the Q factor and resonance
frequency of oscillation micrometer and nanometer scale beams”, Phys. Rev. E. Vol.
69, 036307. DOI: 10.1103.
[30] Weibin, Z. and Turner, K., 2006, “Frequency dependent fluid damping of micro/nano
flexural resonators: Experiment, model and analysis”, Sensors and Actuators A doi:
10.1016 / j. sna.2006.06.01.
[31] Farrokhabadi, A., Rach, R., Abadyan, M., 2013, “Modeling the static response and pull-in
instability of CNT nanotweezers under the Coulomb and van der Waals attractions”,
Physica E, Vol. 53, pp 137-145.
[32] Ramezani, A., 2011, “Stability analysis if electrostatic nanotweezers”, Physica E, Vol. 43,
pp 1783-1791.
[33] Akita, S. and Nakayama, Y., 2002, “Manipulation of Nanomaterial by Carbon Nanotube
Nanotweezers in Scanning Probe Microscope”, The Japan Society of Applied Physics.
Vol. 41, pp. 4242-4245.
[34] Gusso, A., Delben, G.J., 2007, “Influence of the Casimir force on the pull-in parameters of
silicon based electrostatic torsional actuatoars,” Sensors and Actuators, Vol. 135, pp 792-800.
[35] Boggild, P., Hansen, T. M., Tanasa, C., and Grey, F., 2001, “fabrication and actuation of
customized nanotweezers with a 25 nm gap”, Nanotechnology, Vol. 12, pp 331-335.

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BIOGRAPHICAL SKETCH
Bin Liu was born in Huaiyang, Henan, China on Jan 06, 1987. He attended Huocheng
High School in Huocheng, Xinjiang, graduated in 2007. After he received his bachelor degree in
Mechanical Engineering from Xinjiang University located in Urumqi, China in June, 2010, he
joined Tong-Jie Innovation Company as a chassis designer. Two years later, he moved to
Shanghai Automotive Company as a product Engineer for automotive parts design and
manufacturing. He attended the University of Texas – Rio Grande Valley, Edinburg, TX,
receiving a Master of Science in Mechanical Engineering in May, 2016. As a graduate student,
from Fall 2014 he worked as a Graduate Assistant with Dr. Caruntu in the Mechanical
Engineering Department at UTRGV. He currently resides at 1111 W. 23rd
St., Mission, TX
78574.